Invariants
| Level: | $12$ | $\SL_2$-level: | $12$ | Newform level: | $24$ | ||
| Index: | $36$ | $\PSL_2$-index: | $36$ | ||||
| Genus: | $1 = 1 + \frac{ 36 }{12} - \frac{ 4 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$ | ||||||
| Cusps: | $4$ (none of which are rational) | Cusp widths | $6^{2}\cdot12^{2}$ | Cusp orbits | $2^{2}$ | ||
| Elliptic points: | $4$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 12L1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 12.36.1.68 |
Level structure
| $\GL_2(\Z/12\Z)$-generators: | $\begin{bmatrix}7&11\\8&5\end{bmatrix}$, $\begin{bmatrix}11&0\\0&1\end{bmatrix}$, $\begin{bmatrix}11&2\\10&11\end{bmatrix}$ |
| $\GL_2(\Z/12\Z)$-subgroup: | $(C_2\times C_4):\SD_{16}$ |
| Contains $-I$: | yes |
| Quadratic refinements: | none in database |
| Cyclic 12-isogeny field degree: | $8$ |
| Cyclic 12-torsion field degree: | $32$ |
| Full 12-torsion field degree: | $128$ |
Jacobian
| Conductor: | $2^{3}\cdot3$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 24.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 6 x^{2} + z w $ |
| $=$ | $3 y^{2} - 4 z^{2} - 2 z w - w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{4} - 3 x^{2} z^{2} - 3 y^{2} z^{2} + 9 z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ | $=$ | $\displaystyle x$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}y$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{6}w$ |
Maps to other modular curves
$j$-invariant map of degree 36 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -2^6\,\frac{(2z^{3}-w^{3})^{3}}{w^{3}z^{6}}$ |
Modular covers
Cover information
Click on a modular curve in the diagram to see information about it.
|
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.18.0.a.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.18.0.j.1 | $12$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 12.18.1.j.1 | $12$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 12.72.3.g.1 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 12.72.3.bh.1 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 12.72.3.bl.1 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 12.72.3.br.1 | $12$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.dj.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.im.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.jo.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.le.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.3.bbg.1 | $24$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 24.72.3.bbh.1 | $24$ | $2$ | $2$ | $3$ | $1$ | $1^{2}$ |
| 24.72.5.ic.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.72.5.id.1 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 36.108.5.i.1 | $36$ | $3$ | $3$ | $5$ | $1$ | $1^{4}$ |
| 36.324.21.d.1 | $36$ | $9$ | $9$ | $21$ | $11$ | $1^{8}\cdot2^{6}$ |
| 60.72.3.sd.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.72.3.sf.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.72.3.sl.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.72.3.sn.1 | $60$ | $2$ | $2$ | $3$ | $0$ | $1^{2}$ |
| 60.180.13.mh.1 | $60$ | $5$ | $5$ | $13$ | $8$ | $1^{12}$ |
| 60.216.13.ou.1 | $60$ | $6$ | $6$ | $13$ | $3$ | $1^{12}$ |
| 60.360.25.cbf.1 | $60$ | $10$ | $10$ | $25$ | $15$ | $1^{24}$ |
| 84.72.3.oj.1 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.72.3.ol.1 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.72.3.or.1 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.72.3.ot.1 | $84$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 84.288.21.ix.1 | $84$ | $8$ | $8$ | $21$ | $?$ | not computed |
| 120.72.3.eim.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.eja.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.ekq.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.ele.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.gqw.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.3.gqx.1 | $120$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 120.72.5.bgo.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 120.72.5.bgp.1 | $120$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 132.72.3.oj.1 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.72.3.ol.1 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.72.3.or.1 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 132.72.3.ot.1 | $132$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.72.3.oj.1 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.72.3.ol.1 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.72.3.or.1 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 156.72.3.ot.1 | $156$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.dwu.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.dxi.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.dyy.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.dzm.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.fts.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.3.ftt.1 | $168$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 168.72.5.ri.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 168.72.5.rj.1 | $168$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 204.72.3.oj.1 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 204.72.3.ol.1 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 204.72.3.or.1 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 204.72.3.ot.1 | $204$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.72.3.oj.1 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.72.3.ol.1 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.72.3.or.1 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 228.72.3.ot.1 | $228$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.dwu.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.dxi.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.dyy.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.dzm.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.fts.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.3.ftt.1 | $264$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 264.72.5.ri.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 264.72.5.rj.1 | $264$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 276.72.3.oj.1 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 276.72.3.ol.1 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 276.72.3.or.1 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 276.72.3.ot.1 | $276$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.dwu.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.dxi.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.dyy.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.dzm.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.fts.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.3.ftt.1 | $312$ | $2$ | $2$ | $3$ | $?$ | not computed |
| 312.72.5.ri.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |
| 312.72.5.rj.1 | $312$ | $2$ | $2$ | $5$ | $?$ | not computed |